We will derive the necessary twisted joint moment formulae from formulae for twisted moments of \(\zeta (s)\)  with small shifts off of the critical line.
Fortunately there are many known formulae for computing twisted moments of \(\zeta \)  due to connections with the proportion of zeros of \(\zeta \)  lying on the critical line [1]}, [2]}.
Then following work of Young [3]}, we can differentiate these formulae with respect to the shifts to obtained the desired twisted joint moments.
The formula in [3]} is valid for Dirichlet polynomials of length \(T^{1/2 - \varepsilon }\) , and we note that work of Bettin Chandee and Radziwiłł [5]} provides asymptotics for the twisted second moment without shifts for any Dirichlet polynomial of length at most \(T^{17/33- \varepsilon }\) . The twisted fourth moment formula we use was first proven by Hughes and Young [6]} for Dirichlet polynomials of length at most \(T^{1/11 - \varepsilon }\) , which was later increased to \(T^{1/4 - \varepsilon }\)  by Bettin Bui Li and Radziwiłł [7]}.
